São Paulo (CNN)--Brazil is pulling ahead in the race that no one wants to win.

Over the past month, the South American nation has blown past a series of grisly milestones, repeatedly setting new records for most Covid-19 deaths per day. In the past week, it set another record: 12,818 new deaths and more than 464,000 new cases, according to Johns Hopkins University figures -- signs of a viral spread outpacing even that of the United States, the only country in the world harder hit by the pandemic in absolute numbers.

On the outskirts of São Paulo, the crisis is turning things upside down for the small Dr. Akira Tada Emergency Hospital. In normal times, doctors there would stabilize critically ill patients and then send them on to bigger and better-equipped hospitals with intensive care units (ICUs).

But these are not normal times. Few hospitals today have space to take in new patients, even in Brazil's richest and most populous state.

When Dineia Martins Firmino entered the hospital at the beginning of March, doctors intubated the 74-year-old, and told her family that she desperately needed to be moved to an ICU for more sophisticated treatment, according to her granddaughter Pamela Rivitti, 30.

She never made it off the official, government-run list for transfer. "No vacancy appeared at the time she needed it and she ended up dying on Saturday," said Rivitti. "We did the funeral on Sunday."